NATIONAL TAIWAN UNIVERSITY, COLLOQUIUM, MARCH 10, 2015 Quantum and classical annealing in spin glasses and quantum computing Anders W Sandvik, Boston University Cheng-Wei Liu (BU) Anatoli Polkovnikov (BU) C.-W. Liu, A. Polkovnikov, A. W. Sandvik, arXiv:1409.7192 Tuesday, March 10, 15 1
Outline Classical (thermal) fluctuations versus Quantum fluctuations (tunneling) • Computational studies of model systems (spin glasses) • Relevance for adiabatic quantum computing • Monte Carlo simulations and simulated annealing • Quantum annealing for quantum computing • Classical and quantum spin glasses • Dynamical critical scaling Tuesday, March 10, 15 2
Monte Carlo Simulations Example: Particles with hard and soft cores (2 dim) P ( { r i } ) ∝ e − E/T , X E = V ( r i − r j ) r 1 ,r 2 What happens when the temperature is lowered ? Tuesday, March 10, 15 3
Monte Carlo Simulations Transition into liquid state has taken place Slow movement & growth of droplets Is there a better way to reach equilibrium at low T? Tuesday, March 10, 15 4
Simulated Annealing Annealing: Removal of crystal defects by heating followed by slow cooling Simulated Annealing: MC simulation with slowly decreasing T - Can help to reach equilibrium faster Optimization method: express optimization of many parameters as minimization of a cost function, treat as energy in MC simulation Similar scheme in quantum mechanics? Tuesday, March 10, 15 5
Quantum Annealing Reduce quantum fluctuations as a function of time - start with a simple quantum Hamiltonian (s=0) - end with a complicated classical potential (s=1) H ( s ) = sH classical + (1 − s ) H quantum s = s ( t ) = vt, v = 1 /t max H quantum = − ~ 2 d 2 H classical = V ( x ) dx 2 2 m Adiabatic Theorem: If the velocity v is small enough the system stays in the ground state of H[s(t)] at all times At t=t max we then know the minimum of V(x): Ψ ( x ) = δ ( x − x 0 ) Can quantum annealing be more efficient than thermal annealing? Ray, Chakrabarty,Chakrabarty (PRB 1989), Kadowaki, Nishimory (PRE 1998),... Useful paradigm for quantum computing? Tuesday, March 10, 15 6
Quantum Annealing & Quantum Computing D-wave “quantum annealer”; 512 flux q-bits - Claimed to solve some hard optimization problems - Is it really doing quantum annealing? - Is quantum annealing really better than simulated annealing (on a classical computer)? Hamiltonian implemented in D-wave quantum annealer.... Tuesday, March 10, 15 7
Spin Glasses Ising models with frustrated interactions J 34 =+1 N N J 41 =-1 J 23 =-1 X X J ij σ z i σ z σ z i ∈ { − 1 , +1 } H = j , i =1 j =1 J 12 =-1 Hard to find ground states if the interactions J 34 =+1 are highly frustrated (spin glass phase) J 41 =-1 J 23 =-1 - many states with same or almost same energy J 12 =-1 Many (almost all) optimization problems can be mapped onto some general model - hard problems correspond to spin glass physics Quantum fluctuations (quantum spin glasses) The D-wave - add transversal field Ising (H → H + H quantum ) machine is based on this N N model on a X X ( σ + σ x H quantum = − h i = − h i + σ − i ) special lattice i =1 i =1 Nature of ground states of H depends on h and {J ij } Tuesday, March 10, 15 8
Quantum Phase Transition There must be a quantum phase transition in the system H ( s ) = sH classical + (1 − s ) H quantum Ground state changes qualitatively as s changes - trivial (easy to prepare) for s=0 - complex (solution of hard optimization problem) at s=1 → expect a quantum phase transition at some s=s c Simple example: 1D transverse-field Ising ferromagnet N N X X σ z i σ z σ x ( N → ∞ ) h = − s i +1 − (1 − s ) i i =1 i =1 - trivial x-oriented ferromagnet at s=0 ( →→→ ) - z-oriented ( ↑↑↑ or ↓↓↓ , symmetry broken) at s=1 - s c =1/2 (exact solution, mapping to free fermions) Have to pass through s c and beyond adiabatically - how long does it take? s = s ( t ) = vt, v = 1 /t max Let’s look at a simpler problem first... Tuesday, March 10, 15 9
Landau-Zener Problem Single spin in magnetic field, with mixing term H = − h � z − ✏� x = − h � z − ✏ ( � + + � − ) Eigen energies are 1.0 ↑ p ↓ h 2 + ✏ 2 E = ± Smallest gap: Δ =2 ε 0.5 Time-evolution: l ∆ 0.0 E h ( t ) = − h 0 + vt ↓ ↑ To stay adiabatic -0.5 when crossing h=0, the velocity must be -1.0 -1 -0.5 0 0.5 1 v < ∆ 2 (time > ∆ − 2 ) h Suggests the smallest gap is important in general - but states above the gap play role in many-body system What can we expect at a quantum phase transition? Tuesday, March 10, 15 10
Dynamic Critical Exponent and Gap Dynamic exponent z at a phase transition - relates time and length scales At a continuous transition (classical or quantum): - large (divergent) correlation length δ = distance from critical ξ t ∼ ξ z r ∼ | δ | − ν z ξ r ∼ | δ | − ν , point (in T or other param) Continuous quantum phase transition - excitation gap at the transition order parameter (a) depends on the system size and z as ∆ ∼ 1 1 ( N = L d ) L z = N z/d , T c [ g c ] T [ g ] Exponentially small gap at a first-order order parameter (b) (discontinuous) transition ∆ ∼ e − aL Important issue for quantum annealing! T c [ g c ] T [ g ] P. Young et al. (PRL 2008) Exactly how does z enter in the adiabatic criterion? Tuesday, March 10, 15 11
Kibble-Zurek Velocity and Scaling The adiabatic criterion for passing through a continuous phase transition involves more than z Must have v < v KZ , with Kibble 1978 v KZ ∼ L − ( z +1 / ν ) - defects in early universe Zurek 1981 Same criterion for classical - classical phase transitions and quantum phase transitions Polkovnikov 2005 - adiabatic (quantum) - quantum phase transitions - quasi-static (classical) Generalized finite-size scaling hypothesis A ( δ , v, L ) = L − κ / ν g ( δ L 1 / ν , vL z +1 / ν ) ν 0 = d ν , z = z/d A ( δ , v, N ) = N � κ / ν 0 g ( δ N 1 / ν 0 , vN z 0 +1 / ν 0 ) , Will use for spin glasses of interest in quantum computing Apply to well-understood clean system first... Tuesday, March 10, 15 12
Fast and Slow Classical Ising Dynamics Repeat many times, collect averages, analyze,.... Tuesday, March 10, 15 13
Velocity Scaling, 2D Ising Model Repeat process many times, average data for T=T c h m 2 ( δ = 0 , v, L ) i = L − 2 β / ν f ( vL z +1 / ν ) Used known 2D Ising exponents 0 10 L = 128 β =1/8, ν =1 -2 10 L = 12 Adjusted z for -1 L = 24 10 optimal scaling L = 48 -3 10 2 β / ν L = 64 4 5 6 10 10 10 collapse L = 96 2 > L -2 10 L = 128 Result: z ≈ 2.17 <m L = 192 L = 256 consistent with -3 L = 500 10 values obtained L = 1024 polynomial fit in other ways -4 power-law fit 10 Liu, Polkovnikov, 0 2 4 6 8 10 10 10 10 10 Sandvik, PRB 2014 z+1/ ν v L Can we do something like this for quantum models? Tuesday, March 10, 15 14
Quantum Evolution in Imaginary Time Schrödinger dynamic at imaginary time t=-i τ | Ψ ( τ ) i = U ( τ , τ 0 ) | Ψ ( τ 0 ) i Time evolution operator Z τ � d τ 0 H [ s ( τ 0 )] U ( τ , τ 0 ) = T τ exp − τ 0 Dynamical exponent z same as in real time! (DeGrandi, Polkovnikov, Sandvik, PRB2011) • Can be implemented in quantum Monte Carlo Z τ Z τ n Z τ 2 ∞ X | Ψ ( τ ) i = d τ n − 1 · · · d τ 1 [ � H ( τ n )] · · · [ � H ( τ 1 )] | Ψ (0) i d τ n τ 0 τ 0 τ 0 n =0 Simpler scheme: evolve with just a H-product (Liu, Polkovnikov, Sandvik, PRB2013) ∆ s = s M | Ψ ( s M ) i = H ( s M ) · · · H ( s 2 ) H ( s 1 ) | Ψ (0) i , s i = i ∆ s , M How does this method work? Tuesday, March 10, 15 15
QMC Algorithm Illustration Transverse-field Ising model: 2 types of operators: Represented as “vertices” H 1 ( i ) = − (1 − s )( σ + i + σ − i ) H 2 ( i, j ) = − s ( σ z i σ z j + 1) MC sampling of networks of vertices N = 4 M = 7 1 2 3 4 5 6 7 7 6 5 4 3 2 1 h Ψ (0) | H ( s 1 ) · · · H ( s 7 ) | H ( s 7 ) · · · H ( s 1 ) | Ψ (0) i Similar to ground-state projector QMC How to define (imaginary) time in this method? Tuesday, March 10, 15 16
Time and velocity Definitions The parameter in H changes as ∆ s = s M s i = i ∆ s , M Time unit is ∝ 1/N, velocity is v ∝ N ∆ s Def reproduces v-dependence in imag-time Schrödinger dynamics to order v (enough for scaling) To this order we can use “asymmetric” expectation values M k 1 1 M k Y Y Y h A i k = h Ψ (0) | H ( s i ) H ( s i ) A H ( s i ) | Ψ (0) i Y Y Y h A i k = h Ψ (0) | H ( s i ) H ( s i ) A H ( s i ) | Ψ (0) i i =1 i = M i = k i = M i =1 i = k All s in one simulation! Collect data, do scaling analysis... Tuesday, March 10, 15 17
2D Transverse-Ising, Scaling Example A ( δ , v, L ) = L − κ / ν g ( δ L 1 / ν , vL z +1 / ν ) If z, ν known, s c not: use vL z +1 / ν = constant 1.0 0.9 for 1-parameter scaling 0.8 0.8 Example: Binder cumulant 0.6 L = 12 U 0.24 0.25 L = 16 L = 32 0.4 L = 48 L = 56 L = 60 Should have step from 0.2 U=0 to U=1 at s c - crossing points for 0.0 0.15 0.20 0.25 0.30 finite system size S Do similar studies for quantum spin glasses Tuesday, March 10, 15 18
Recommend
More recommend
